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I have a small application of the central limit theorem:

"Take a sample of 40 exponentially distributed random numbers and calculate their mean. Repeat 1000 times, record each measured mean and calculate the mean of those 1000 means."

Here's my Rcode:

nsim   = 1000
nsamp  = 40
lambda = 0.2

clt <- data.frame(m=numeric(nsim), s=numeric(nsim))

for(s in 1:nsim)
{
    clt$v[s] <- mean(rexp(nsamp, lambda))
    clt$s[s] <- sd(rexp(nsamp, lambda))
}   

cat(mean(clt$v), "\n")
    cat(mean(clt$s), "\n")

I can observe that with increasing number of simulations (i.e. increasing the repetitions of the sampling event), the mean of means and mean of variance more closely agree with the theoretical values (1/lambda).

What I wonder is, does the Central-limit theorem only depend on the number of sampling events (1000) or is it just as much dependent on sufficiently large sample size (i.e. 40 in this example)?

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    $\begingroup$ What, in you own words, does the CLT actually say? You might enjoy reading some of our higher-voted threads on the CLT. $\endgroup$
    – whuber
    Jan 18 '15 at 16:34
  • $\begingroup$ I don't see the Central Limit Theorem here, only the Law of Large Numbers. $\endgroup$ Feb 22 '17 at 12:27
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The central limit theorem states that you should take samples of sufficiently large size (40 in your example), but it does not provide a limit on the minimum number of such samples, i.e., the statement holds regardless of the number of samples. See point #1 in this answer.

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The CLT is generally considered a theoretical distribution. We don't actually generate 1000 (or 10,000 or 100,000) samples from it. Rather, we imagine what would happen if we were to replicate this experiment 1000 (or, technically, infinite) times.

The R code you post caps it at 1000, probably because by 1000 samples it will demonstrate the properties of the CLT. But, again, the CLT really says what would happen if you repeated it an infinite number of times.

So, to your question:

does the Central-limit theorem only depend on the number of sampling events (1000) or is it just as much dependent on sufficiently large sample size (i.e. 40 in this example)?

It depends on both (kind of). The number of sampling events actually is infinite (though the CLT properties hold with smaller distributions), and as the sample size increases, the more normal (and smaller the variance) the distribution of means becomes.

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